NEWTON, ISAAC (b. Woolsthorpe, England, 25 December 1642; d. London, England, 20 March 1727), mathematics, dynamics, celestial mechanics, astronomy, optics, natural philosophy.

    inertia and Kepler's law of areas (generalized to hold for an arbitrary central orbit).

Combining proposition 1 and proposition 2, Newton showed the physical significance of the law of areas as a necessary and sufficient condition for a central force (supposing that such forces exist; the “reality” of accelerative and motive forces of attraction is discussed in book III). In proposition 3, Newton dealt with the case of a body moving around a moving, rather than a stationary, center. Proposition 4 is concerned with uniform circular motion, in which the forces (F, f) are shown not only to be directed to the centers of the circles, but also to be to each other “as the squares of the arcs [S, s] described in equal times divided respectively by the radii [R, r] of the circles” (F:f = S/R2:s/r2). A series of corollaries demonstrate that F:f = V2/R:v2r = R/T2:r/t2, where V, v are the tangential velocities, and so on; and that, universally, T being the period of revolution, if T?Rn, V?1/Rn-1, then F?1/R2n-1, and conversely. A special case of the last condition (corollary 6) is T?R3/2, yielding F?1/R2, a condition (according to a scholium) obtaining “in the celestial bodies,” as Wren, Hooke, and Halley “have severally observed.” Newton further referred to Huygens' derivation, in De horologio oscillatorio, of the magnitude of “the centrifugal force of revolving bodies” and introduced his own independent method for determining the centrifugal force in uniform circular motion. In proposition 6 he went on to a general concept of instantaneous measure of a force, for a body revolving in any curve about a fixed center of force. He then applied this measure, developed as a limit in several forms, in a number of major examples, among them proposition 11.

The last propositions of section 2 were altered in successive editions. In them Newton discussed the laws of force related to motion in a given circle and equiangular (logarithmic) spiral. In proposition 10 Newton took up elliptical motion in which the force tends toward the center of the ellipse. A necessary and sufficient cause of this motion is that “the force is as the distance.” Hence if the center is “removed to an infinite distance,” the ellipse “degenerates into a parabola,” and the force will be constant, yielding “Galileo's theorem” concerning projectile motion.

Section 3 of book I opens with proposition 11, “If a body revolves in an ellipse; it is required to find the law of the centripetal force tending to the focus of the ellipse.” The law is: “the centripetal force is inversely ... as the square of the distance.” Propositions 12 and 13 show that a hyperbolic and a parabolic orbit imply the same law of force to a focus. It is obvious that the converse condition, that the centripetal force varies inversely as the square of the distance, does not by itself specify which conic section will constitute the orbit. Proposition 15 demonstrates that in ellipses “the periodic times are as the 3/2th power of their greater axes” (Kepler's third law). Hence the periodic times in all ellipses with equal major axes are equal to one another, and equal to the periodic time in a circle of which the diameter is equal to the greater axis of each ellipse. In proposition 17, Newton supposed a centripetal force “inversely proportional to the squares of the distances” and exhibited the conditions for an orbit in the shape of an ellipse, parabola, or hyperbola. Sections 4 and 5, on conic sections, are purely mathematical.

In section 6, Newton discussed Kepler's problem, introducing methods of approximation to find the future position of a body on an ellipse, according to the law of areas; it is here that one finds the method of successive iteration. In section 7, Newton found the rectilinear distance through which a body falls freely in any given time under the action of a “centripetal force ... inversely proportional to the square of the distance ... from the centre.” Having found the times of descent of such a body, he then applied his results to the problem of parabolic motion and the motion of “a body projected upwards or downwards,” under conditions in which “the centripetal force is proportional to the ... distance.” Eventually, in proposition 39, Newton postulated “a centripetal force of any kind” and found both the velocity at any point to which any body may ascend or descend in a straight line and the time it would take the body to get there. In this proposition, as in many in section 8, he added the condition of “granting the quadratures of curvilinear figures,” referring to his then unpublished methods of integration (printed for the first time in the De quadratura of 1704).

In section 8, Newton often assumed such quadrature. In proposition 41 he postulated “a centripetal force of any kind”; that is, as he added in proposition 42, he supposed “the centripetal force to vary in its recess from the center according to some law, which anyone may imagine at pleasure, but [which] at equal distances from the centre [is taken] to be everywhere the same.” Under these general conditions, Newton determined both “the curves in which bodies will move” and “the times of their motions in the curves found.” In other words, Newton presented to his readers a truly general resolution of the inverse problem of finding the orbit from a given law of force. He extended this problem into a dynamics